Lacunary statistical convergence

Abstract

The sequence x is statistically convergent to L provided that for each ε < 0, {the number of k ≤ n: |xk - L| ≥ ε} = 0. In this paper we study a related concept of convergence in which the set {k: k ≤ n) is replaced by {k: kr-1 < k ≤ kr}, for some lacunary sequence {kr} . The resulting summability method is compared to statistical convergence and other summability methods, and questions of uniqueness of the limit value are considered. © 1993 by Pacific Journal of Mathematics.